Slope Stability Theory

ENVIRONMENTAL GEOTECHNICS Slope Stability Theory Prof. Ing. Marco Favaretti University of Padova Department of Civil, Environmental and Architectural Engineering, Via Ognissanti, 39 Padova (Italy) phone: +39.049.827.7901 e-mail: marco.favaretti@unipd.it website: www.marcofavaretti.net 1

Introduction to SLOPE STABILITY Landslides are the result of slope instability. It occurs in many parts of the world, especially in those areas with problematic soils/rocks and/or adverse environmental conditions. They are usually caused by excavation, undercutting the foot of an existing slope, improper surface and subsurface drainage systems, tunnel collapse of underground caverns, surface and subsurface erosion, or by a shock caused by earthquake or blasting, which liquefies the soil. 2

Introduction to SLOPE STABILITY In analyzing the landslide problem, engineers and geologists often look at it from different points of view. The geologist regards a landslide as one of many natural processes acting as part of the geological cycle. They are interested only in the ground movement with respect to the geological and hydrological features. The geotechnical engineer is interested in the soil types, their engineering behaviour, the maximum height of the slope, and maximum slope angle in terms of a safety factor. In most cases, they do not understand the geological formation and environmental factors that cause a landslide. 3

Introduction to SLOPE STABILITY Even within the engineering group there are different perspectives. The practitioner is interested in the measurements of soil-rock properties, ground movements, and local environmental conditions to design a solution; the theoretician is interested in idealizing the failure surface in order to fit it into a mathematical description for use in subsequent efforts to model the system. Since the landslide problem is not a simple matter, it requires knowledge from other disciplines. Therefore, a joint effort from geologists, geotechnical engineers, and seismologists is required to tackle this problem. 4

FACTORS AFFECTING SLOPE INSTABILITY Factors affecting earth slope instability are: (a) external loading conditions including surcharge loading, earthquake actions, blasting vibration, moving vehicle, and construction operation; (b) environmental factors including rainstorms hurricane (typhoon), flash flood, El Niño and La Niña effects, dry-wet and freeze-thaw cycles, acid rain and acid drainage, pollution intrusion, treevegetation roots, and animal, insect, and microbiological attack. 5

FACTORS AFFECTING SLOPE INSTABILITY The EXTERNAL LOADING conditions can be divided into two groups: dead load environmental load The dead load, in general, is also called a surcharge load, which is also called a static load. The environmental loads are mostly dynamic in nature and can be violent, such as earthquake and blasting vibrations. The INTERNAL FACTORS include volume changes, shrinkage and swelling, and surface and internal cracking of soil mass, which consequently changes bond stress between soil particles and loss shear strength, bearing capacity, etc. 6

Definition of LANDSLIDE The downward falling or sliding of a mass of soil, detritus, or rock on or from a steep slope. Introduction to SLOPE STABILITY Slope stability problems have been faced throughout history when men or nature has disrupted the delicate balance of natural soil slopes. Need to understand analytical methods, investigative tools, and stabilization methods to solve slope stability problems. 7

Aims of Slope Stability Analysis The primary purpose of slope stability analysis is to contribute to the safe and economic design of excavations, embankments, earth dams, landfills, and spoil heaps. The aims of slope stability analyses are: (1) To understand the development and form of natural slopes and the processes responsible for different natural features. (2) To assess the stability of slopes under short-term (often during construction) and long-term conditions. 8

AIMS OF SLOPE STABILITY ANALYSIS (3) To assess the possibility of landslides involving natural or existing engineered slopes. (4) To analyze landslides and to understand failure mechanisms and the influence of environmental factors. (5) To enable the redesign of failed slopes and the planning and design of preventive and remedial measures, where necessary. (6) To study the effect of seismic loadings on slopes and embankments. 9

NATURAL AND ARTIFICIALS SLOPES The analysis of slopes takes into account a variety of factors relating to topography, geology, and material properties, often relating to whether the slope was naturally formed or engineered. Natural slopes that have been stable for many years may suddenly fail because of changes in topography, seismicity, groundwater flows, loss of strength, stress changes, and weathering. Significant uncertainty exists about the stability of a natural slope. 10

NATURAL AND ARTIFICIALS SLOPES Knowing that old slip surfaces exist in a natural slope makes it easier to understand and predict the slope s behavior. The shearing strength along these slip surfaces is often very low because prior movement has caused slide resistance to peak and gradually reduce to residual values. Engineered slopes may be considered in three main categories: embankments, cut slopes, and retaining walls. As these slopes are man-made less uncertainty exists about their stability. 11

FACTORS INFLUENCING SLOPE STABILITY The main items required to evaluate the stability of a slope are: A. Shear strength of the soils B. Slope geometry C. Pore pressures or seepage forces D. Loading and environmental conditions (1) Shear strengths should be provided as undrained strength, cu (in some figures or equations s u ) or as drained parameters, c and φ. (2) Slope geometry may be known for existing, natural slopes or may be a design parameter for embankments and cut slopes. (3) A major contributor to many slope failures is the change in effective stress σ caused by pore water pressures u. These tend to alter the shear strength of the soil along the shear zone. 12

B-H B H 13

FACTOR OF SAFETY (FOS) CONCEPTS Function of the FOS: to account for uncertainty, and thus to guard against ignorance about the reliability of the analysis input data, such as, strength parameters, pore pressure distribution, and stratigraphy. The lower the quality of the site investigation, the higher the FOS should be. The FOS used in design will vary with material type and performance requirements. The required FOSs (nonseismic) are usually around 1,30. Higher factors may be required if there is a high risk of loss of life or uncertainty regarding the pertinent design parameters. Likewise, lower FOSs may be used if the engineer is confident of the accuracy of input data and if the construction is being monitored closely. 14

FACTOR OF SAFETY First Definition In most limit equilibrium analyses shear strength required (or mobilized) along a potential failure surface to just maintain stability is calculated and then compared to the magnitude of available shear strength. The FOS is assumed to be constant for the entire failure surface. FOS will be given by the ratio of available to required shear strength: τreq = τ mob τ F = f τ req τ req = c u F for total stresses τ req = c' Fc + σ' tan φ F φ for effective stresses 15

FACTOR OF SAFETY First Definition The adoption of F c and F f allows different proportions of the cohesive (c ) and frictional (φ ) components of strength to be mobilized along the failure surface. However, most limit equilibrium methods assume F c and F f, implying that the same proportion of the c and φ components are mobilized at the same time along the shear failure surface. 16

FACTOR OF SAFETY Second Definition Another definition of FOS often considered is the ratio of total resisting forces to total disturbing (or driving) forces for planar failure surfaces or the ratio of total resisting to disturbing moments, as in the case for circular slip surfaces. Realize that these different values of the FOS obtained using the three methods, that is, mobilized strength, ratio of forces, or ratio of moments, will not give identical values for c φ soils. 17

FACTOR OF SAFETY First definition s u = c u τreq = τ mob Second definition 18

PORE WATER PRESSURES If an effective stress analysis is to be performed, pore water pressures will have to be estimated at relevant locations in the slope. These pore pressures are usually estimated from groundwater conditions that may be specified by one of the following methods: (1) Phreatic Surface: This surface, or line in two dimensions, is defined by the free groundwater level. Delineated, in the field, by using open standpipes as monitoring wells. (2) Piezometric Data: Specification of pore pressures at discrete points, within the slope, and use of an interpolation scheme to estimate the required pore water pressures at any location. Determined from field piezometers, a manually prepared flow net or a numerical solution using finite differences or finite elements. 19

PORE WATER PRESSURES (3) Pore Water Pressure Ratio: This is a popular and simple method for normalizing pore water pressures measured in a slope according to the definition: r u = u σ v where u is the pore pressure and σ v, is the total vertical subsurface soil stress at depth z. Effectively, the r u, value is the ratio between the pore pressure and the total vertical stress at the same depth. This factor is easily implemented, but the major difficulty is associated with the assignment of the parameter to different parts of the slope. It is usually reserved for estimating the FOS value from slope stability charts or for assessing the stability of a single surface. 20

PORE WATER PRESSURES (4) Piezometric Surface is defined for the analysis of a unique, single failure surface. This approach is often used for the back analysis of failed slopes. The piezometric surface may be not overlapped with the phreatic surface, as the calculated pore water pressures will be different for the two cases. 21

PORE WATER PRESSURES Phreatic surface If a phreatic surface is defined, the pore water pressures are calculated for the steady-state seepage conditions. This concept is based on the assumption that all equipotential lines are straight and perpendicular to the segment of the phreatic surface passing through a slice-element in the slope. 22

PORE WATER PRESSURES Phreatic surface Thus if the inclination of the phreatic surface segment is θ, and the vertical distance between the base of the slice and the phreatic surface is h, the pore pressure is given by: u = γ w ( 2 ) h cos θ w This is a reasonable assumption for a sloping straight-line phreatic surface, but will provide higher or lower estimates of pore water pressure for a curved (convex) phreatic surface. 23

PORE WATER PRESSURES Piezometric surface Note that the vertical distance (elevational head) is taken to represent the pressure. There are some computer programs that oversimplify the analysis by misinterpreting a phreatic surface as a piezometric surface. With this erroneous assumption, the overestimated pore pressure head is incorrectly taken as the vertical distance between the phreatic surface and base of slice. 24

LIMIT EQUILIBRIUM METHOD L.E.M. There are numerous methods currently available for performing slope stability analyses. The majority of these may be categorized as limit equilibrium methods. All the procedures currently used are based on the L.E.M. and have the following assumptions in common: Mohr-Coulomb s failure criterion is satisfied along the assumed failure surface, which may be a straight line, circular arc, logarithmic spiral, or other irregular surface. Plain strain conditions are supposed. The actual strength of the soil is compared with the value required for the equilibrium of the soil mass and this ratio is a measure of the safety factor. 25

LIMIT EQUILIBRIUM METHOD L.E.M. In this method soils are treated as rigid-plastic materials and due to this assumption the analysis does not consider deformations. Therefore, this method allows to only condition at the onset of failure. These methods are very similar to the kinematic approach, but frequently the restrictions of a kinematically admissible mechanisms are ignored. Limit Equilibrium Methods can be subdivided in two principal categories: Methods that consider only the whole free body (Culmann method, friction circle method). Methods that divide the free body into many vertical slices and consider equilibrium of each slide (method of slices). 26

LIMIT EQUILIBRIUM METHOD L.E.M. 27

Slope Instability Instability of natural slopes (generally the product of erosion) or artificial slopes (created by cuttings, excavations or the building of embankments for instance) results in most cases of a combination of gravitational forces and water pressures. When it occurs failure involves a mass of soil within which the forces due to gravity (i.e. the weight of soil mass) are out of balance. The presence of water increases the imbalance since it plays the role of a lubricant at the surface along which the soil mass is moving, gradually increasing its velocity and leading ultimately to total failure. Failure mechanisms depends on the nature and state of soil. 28

Slope Instability translational slip: the failure surface is parallel to the slope and the movement of the failing soil mass consists mainly of a translation. 29

Slope Instability circular slip surface: failure mainly occurs through rotation of the unstable volume of soil. The position of the centre of rotation plays an important role in the stability analysis of such slopes. 30

Slope Instability In both cases failure occurs (theoretically) once the driving shear forces, due to the weight of the moving soil mass, become equal (or larger) to the resisting shear forces of soil. If shear stress mobilized by the failing soil mass at its base is τ mob and if shear strength of soil is referred to as τ f then the factor of safety F against shear failure is simply defined as: F τ τ f = 01 mob τreq = τ mob 31

Slope Instability Failure occurs (in theory) when F 1. In practice failure is related more to the velocity with which the failing block of soil is moving; experimental evidence shows that failures do occur for slopes with factors of safety of up to 1.25. So it is tempting to assume that slopes with a factor of safety F 1.25 are stable. However, such an assumption is only realistic as long as parameters representative of soil behaviour are used in the calculation. Several instances of slope failure occurring with a factor of safety of 1.5 or more were reported in the literature. In all cases, wrong shear strength values (usually overestimates) were used in the calculations. 32

Slope Instability Potential problems related to the determination of soil parameters based on laboratory measurements. Undrained shear strength c u of a (stiff) clay, measured in the laboratory on a small size sample, often represents an overestimate of the actual c u value in situ, because it is unlikely that a small sample would contain fissures that characterize a thick stiff clay layer in the field. It is therefore essential to bear in mind that the outcome of any design method depends entirely on the soil parameters used. The designer must master design techniques, but equally important, he must be careful when selecting reliable values of relevant soil parameters. 33

Slope Instability Accordingly, the selection of appropriate and reliable soil strength parameters is not an option, but rather a precondition to a safe design. This selection depends on the state of the soil (O.C./N.C., dense/loose), on drainage conditions, and on failure criteria. Thus, if failure is predicted to occur under drained conditions along an old slip surface, then the residual angle of shearing resistance will be most appropriate for the design, and the soil shear strength is calculated as follows: f ' n ' r τ = σ tanφ 02 34

Slope Instability If failure would occur within a soil mass that has never experienced failure in the past then, a priori, it would be appropriate to select the peak angle of shearing resistance in conjunction with O.C. or dense soils, or the critical angle if the soil is N.C. or loose. However, in practice, it is advisable to use the critical angle, so that for N.C. or lightly O.C. clays, as well as for (dense and loose) sands, the drained shear strength is estimated from equation: For heavily O.C. clays, the drained shear strength might be affected by the apparent cohesion, so that: τ τ f f = σ ' n = c' +σ tanφ ' n ' c tanφ ' c 03 04 35

Slope Instability For slow draining soils (mainly clays), the excess pore water pressure generated at the onset of loading may create conditions whereby failure can take place in the short term (that is before the occurrence of any significant dissipation of excess pore water pressure). In which case, the undrained shear strength c u must be used as in equation: τf = c 05 u 36

Stability of infinite slopes Slopes with a failure mechanism such as the one depicted in figure are often assumed to be of infinite length because of the usually large ratio of slip plane length to its depth. The soil behaviour is modelled to be whereby the potential slip plane, situated at a depth z, is parallel to the ground surface, sloping at an angle β. 37

Stability of infinite slopes Pore water pressure u can be evaluated as follows: u h p γ = 06 w h p : pressure head γ w : unit weight of water 38

Stability of infinite slopes In the case of a purely frictional material (c = 0 φ 0) shear strength at failure is related to the effective normal stress. For an element of soil 1-unit long, 1-unit wide and z units deep the equilibrium requirements are such that: σ ' n + u = W cosβ τ mob = W sinβ 07 08 where τ mob is the mobilized shear stress. The total weight W of the element is calculated as follows: W 09 = 1 1 z γ cosβ γ : total unit weight of soil (i.e. neglecting the difference between bulk and saturated unit weights). 39

Stability of infinite slopes Shear strength of equation (03) can now be expressed as follows: ' [( W cosβ) u] tanφ c W = γ z cosβ ' ' τf = σn tanφ c = 03 Substituting for W (eq.09) into the expression (01), then rearranging, it obtains: ' tanφc = tanβ ' (( W cosβ) u) tanφc (( γ z cosβ cosβ) u) = W sinβ ( γ z cosβ) sin ' τf tanφc F = = = 10 τ β mob u 1 ( γ z cosβ) cosβ ' tanφc = tanβ 1 introducing u from equation (06) it follows that: u γ z cos 2 β ' c tanφ γ w hp F = 1 β 2 11 tan γ z cos β F depends on slope angle β, depth of the slip plane z and pressure head h p. 40

Stability of infinite slopes Consider a general case of flow in a sliding soil mass of thickness z. Water is flowing at an angle α to the horizontal - Soil is assumed to be dry at the top up to a depth z w. If a standpipe is inserted to the depth of the slip plane, the water would rise to the depicted height h p. From the geometry of the figure: h p = ( z z ) w cosβ cosα cos ( ) α β 12 41

Stability of infinite slopes F = tanφ ' c tanβ 1 γ w h p 2 γ z cos β Substituting for h p into equation (11), the general expression of the factor F in the case of a slope of infinite extent in a frictional soil affected by water seepage, is: F = ' tanφ tanβ 1 γ γ cosα cosβ cos α β z w 1 ( ) c w 13 Clearly, the factor of safety depends on the direction of flow. z 42

Stability of infinite slopes Flow parallel to the slope When the angles α and β are identical equation (13) reduces to the following: F = tanφ tanβ ' c w w 14 γ 1 γ z 1 z for a dry slope (z w = z), the factor of safety is: tanφ F = tan β ' c 15 Similarly for a waterlogged slope (z w = 0): Assuming a ratio γ w 1 γ 2 ' tanφ γ F = c 1 w 16 tanβ then equation (14) can be expressed as follows: F = 1 2 ' tanφ tanβ 1 + γ z z c w 17 43

Stability of infinite slopes Flow parallel to the slope When the water is flowing parallel to the slope F is proportional to the ratio (z w / z). The value of F can be determined using a critical angle of shear strength φ c and sloping at an angle β, the value F is read on the appropriate (interpolated if need be) curve, and the factor of safety F is thereafter calculated as follows: F z = F' 1 + z w 18 44

Stability of infinite slopes Horizontal Flow A horizontal flow implies an angle α = 0. Equation (13) reduces to the following: F = tanφ ' c tanβ 1 γ w γ 1 cos 2 β 1 z w z 19 Stability of infinite slopes Vertical Flow The slope is waterlogged (z w = 0). This flow implies an angle α = π/2. Equation (13) reduces to the following: F ' c tanφ = tan β 20 Equations 20 and 15 are identical stability is not affected by water seepage in the case of vertical downward flow 45

Stability of infinite slopes Effects of cohesion For heavily OC clays the drained shear strength is affected by the apparent cohesion. However, the stability analysis is not altered a great deal; the reason being that, while the mobilized shear stress within the soil mass remains the same as defined by equation (08) τ mob = W sinβ 08 the shear strength is now given by equation (04) Substituting for τ f from equation (04) into equation (10), the factor F is as follows: f ' n τ = c' +σ tanφ 04 ' c F = ' τf c' tanφc γ cosα ( ) z = + 1 w 1 w 21 τmob z γ sinβ cosβ tanβ γ cosβ cos α β z 46

Stability of infinite slopes Effects of cohesion A comparison of equations (21) and (13) shows that, for cohesive soils, the factor of safety is simply increased by an amount (2c /z γ sin 2β) as opposed to a granular soil. EXAMPLE a slope inclined at an angle β = 15 to the horizontal, in a soil characterized by a cohesion c = 10 kn/m 2 and a total unit weight γ = 20 kn/m 3 with a 6 m deep slip plane, will have its factor of safety against sliding increased by a respectable value of 0.33, irrespective of the flow direction or the water level within the slope. 47

Stability of infinite slopes The most simple way to analyze slope stability. It is used when: A slope extends for a relatively long distance and has a consistent subsoil profile. The failure plane is parallel to the surface of the slope and the limit equilibrium method can be applied readily. Infinite slopes can be studied under several configurations: Infinite slope in dry sand. Infinite slope in c φ soil with seepage: seepage parallel to the slope. Infinite slope in c φ soil with seepage: horizontal seepage (i.e. rapid dewatering of canals). 48

INFINITE SLOPE ANALYSIS Dry Soil Dry Soil Interslice tensions on the lateral sides of each element are equal, so the soil mass moves like a continuum. The normal (W ) and driving (W // ) forces are determined: W = W cos β = γ b h 1 cos β W // = W sin β = γ b h 1 sin β where β is the constant inclination of the slope 49

INFINITE SLOPE ANALYSIS Dry Soil The available frictional strength along the failure plane will depend on φ and is given by S = c b sec β + N tan φ = c b sec β + W tan φ If we consider the FOS as the ratio of available strength to strength required to maintain stability (limit equilibrium), the FOS will be given by FOS = S W // = c' b secβ + N tanφ' W sinβ = c' b secβ + W cosβ tanφ' W sinβ If we assume c = 0, FOS will be: FOS tanφ = tan β 50

INFINITE SLOPE ANALYSIS Soil with seepage Seepage parallel to the slope If a saturated slope, in cohesive c - φ soil, has seepage parallel to the slope surface the same limit equilibrium concepts may be applied to determine the FOS, which will now depend on the effective normal force (N ). The total and effective weight of the slice, in this case, are respectively given by: W = γ sat b h 1 and W = (γ sat - γ w ) b h 1 = γ b h 1 51

INFINITE SLOPE ANALYSIS Soil with seepage The effective normal force N is given by: N = N U where N is the total normal force and U the pore water force acting on the base of the slice: N = W = W cos β = γ sat b h cosβ ( 2 ) b γ h cos β = γ b h β U = w w cos cosβ So the effective normal force N is given by: N = (γ sat - γ w ) b h 1 cos β = γ b h 1 cos β The driving forces can be given by: W // = W sin β = γ b h 1 sin β P w = p w V = γ w i V = γ w i b h 1 52

INFINITE SLOPE ANALYSIS Soil with seepage where P w is the seepage force and i is the hydraulic gradient defined as the ratio between the hydraulic head measurements over the length of the flow path: h b tanβ i = = = sinβ L b/cosβ So P w will be given by: P w = γ w i b h 1 = γ w b h 1 sin β The available frictional strength along the failure plane will depend on φ and the effective normal force and is given by S = c b sec β + (N U) tan φ = c b sec β + N tan φ = c b sec β + γ b h 1 cos β tan φ 53

INFINITE SLOPE ANALYSIS Cohesionless Soil So the FOS for this case will be FOS = W // S + P w = c' b secβ + γ' b h cosβ tanφ γ' b h sinβ + γ b h sinβ w = c' b secβ + γ' b h cosβ tanφ γ b h sinβ sat For a c = 0 soil, the above expression may be simplified to give FOS = γ' γ sat tanφ' tanβ If we suppose γ γ w, γ sat = γ + γ w 2 γ, so FOS become: FOS = 1 2 tanφ' tanβ 54

INFINITE SLOPE ANALYSIS Soil with seepage Horizontal Seepage The total and effective weight of the slice, also in this case, are respectively given by: W = γ sat b h 1 W = (γ sat - γ w ) b h 1 = γ b h 1 The effective normal force N is given by: N = N U cos β where N is the total normal force and U the pore water force acting on the base of the slice: N = W = W cos β = γ sat b h cos β and U = γ w b h 55

INFINITE SLOPE ANALYSIS Soil with seepage So the effective normal force N is given by: N = (γ sat - γ w ) b h 1 cos β = γ b h 1 cos β The driving forces can be given by: W // = W sin β = γ b h 1 sin β P w = p w V = γ w i V = γ w i b h 1 where P w is the seepage force and i is the hydraulic gradient defined as the ratio between the hydraulic head measurements over the length of the flow path, that, in this case, is given by: i h b tanβ = = = tanβ L b 56

INFINITE SLOPE ANALYSIS Soil with seepage So P w will be given by: P w = γ w i b h 1 = γ w b h 1 tan β In this seepage condition, P w lies on horizontal direction, so we can subdivide P w in the perpendicular and parallel components: P w, = P w sin β = γ w b h 1 tan β sin β P w,// = P w cos β = γ w b h 1 tan β cos β The available frictional strength along the failure plane will depend on φ, the effective normal force N and P w, and is given by S = c b sec β + (N - P w, ) tan φ = c b sec β + (γ b h 1 cos β - γ w b h 1 tan β sin β) tan φ 57

INFINITE SLOPE ANALYSIS Cohesionless Soil Then, the FOS for this case will be If we suppose c = 0 and γ γ w FOS become: 58

INFINITE SLOPE ANALYSIS Comparison An example can show better the differences of the previous determined FOSs. Example: Cohesionless soil: γ = 18 kn/m 3, φ = 30, FOS min = 1,3. Calculate maximum slope angles in all three previous slope configurations. Dry Soil Seepage parallel to the slope Horizontal Seepage It is possible to note that the most onerous configuration is that of horizontal seepage. 59

CIRCULAR SURFACE ANALYSIS Circular failure surfaces are found to be the most critical in slopes consisting of homogeneous materials. There are two analytical methods that may be used to calculate the FOS for a slope : the circular arc (φ u = 0 ) method The friction circle method. 60

CIRCULAR SURFACE ANALYSIS Circular Arc Method The simplest circular analysis is based on the assumption that a rigid, cylindrical block will fail by rotation about its center and that the shear strength along the failure surface is defined by the undrained shear strength. The shear strength angle φ u is assumed to be zero (hence the φ u = 0 method ). The FOS for such a slope may be analyzed by taking the ratio of the resisting and overturning moments about the center of the circular surface O. 61

CIRCULAR SURFACE ANALYSIS Circular Arc Method If the overturning moments are given by (W 1 x 1 ) and resisting moments are given by (c u L R + W 2 x 2 ), the factor of safety FOS for the slope will be: FOS = W 2 x2 + c W x 1 u 1 L R If the undrained shear strength varies along the failure surface, the (c u L) term must be modified and treated as a variable in the above formulation. 62

CIRCULAR SURFACE ANALYSIS Friction Circle Method This method is useful for homogeneous soils with φ > 0 (shear strength depends on normal stress. It may be used when both cohesive and frictional components for shear strength have to be considered in the calculations. The method is equally suitable for total or effective stress types of analysis. The method attempts to satisfy the requirement of complete equilibrium by assuming the direction of the resultant of the normal and frictional component of strength mobilized along the failure surface. This direction corresponds to a line that forms a tangent to the friction circle, with a radius, R f = R sin φ m, where: φ m tanφ = arctan F φ This assumption is guaranteed to give a lower bound FOS value (Lambe, Whitman, 1969). 63

CIRCULAR SURFACE ANALYSIS Friction Circle Method The cohesive shear stresses, acting along the arc ab of the sliding surface, have a resultant, C m, that acts parallel to the direction of the chord ab. Its location may be found by taking moments of the distribution and the resultant, C m, about the circle center. This line of action of resultant, C m, can be located using Distance R c is: m R c = ( ab) ( ab) arc chord R m 64

CIRCULAR SURFACE ANALYSIS Friction Circle Method The actual point of application, A, is located at the intersection of the effective weight force, which is the resultant of the weight and any pore water forces. The resultant of the normal and frictional (shear) force, P, will then be inclined parallel to a line formed by a point of tangency to the friction circle and point A. As the direction of C m is known, the force polygon can be closed to obtain the value of the mobilized cohesive force. The final FOS is computed with the assumption F φ = F c = FOS along the failure circular surface. m m 65

CIRCULAR SURFACE ANALYSIS Friction Circle Method The solution procedure is usually followed graphically. (1) Calculate weight of slide, W (2) Calculate magnitude and direction of the resultant pore water force, U (may need to discretize slide into slices) (1) Calculate perpendicular distance to the line of action of C m (4) Find effective weight resultant, W from m forces W and U, and its intersection with the line of action of C m at A (5) Assume a value of F φ (6) Calculate the mobilized friction angle φ m = tan -1 [(tan φ ) / F φ ] m 66

CIRCULAR SURFACE ANALYSIS Friction Circle Method (7) Draw the friction circle, with radius R f = R sin φ m (8) Draw the force polygon with W appropriately inclined, and passing through point A (9) Draw the direction of P, tangential to the friction circle (10) Draw direction of C m, according to the inclination of the chord linking the end-points of the failure surface (11) The closed polygon will then m provide the value of c m (12) Using this value of C m, calculate F c = c (ab) chord / C m. m (13) Repeat steps 5 to 12 until F c F f. 67

METHOD OF SLICES If the mobilized strength for a c φ soil is to be calculated, the distribution of the effective normal stresses along the failure surface must be known. This condition is usually analyzed by discretizing the mass of the failure slope into smaller slices and treating each individual slice as a unique sliding block. The method of slices is used by most computer programs, as it can readily accommodate complex slope geometries, variable soil conditions, and the influence of external boundary loads. 68

METHOD OF SLICES Basic assumptions (1) There are several formulations of the method of slices in relation to the assumption that the numerous authors had made. We can subdivide them in three categories: 1. Assumptions on interslice forces direction (Bishop, 1955; Spencer, 1967; Morgenstern-Price, 1965) 2. Assumptions on the thrust line position (Janbu, 1954) 3. Assumptions on the interslice forces distribution (Sarma, 1973; Correia, 1988) 69

METHOD OF SLICES Basic assumptions (2) All these methods are based on similar concepts, but they give different FOSs values through the previous assumptions. However, all of them are based on the Mohr-Coulomb failure criterion (rigidperfect plastic behavior). All limit equilibrium methods for slope stability analysis divide a slide mass into n smaller slices, so that we can approximate the irregular base of the slice (often an arc) as a chord. Another hypothesis is that FOS is considered constant along the failure surface. 70

METHOD OF SLICES System of forces Each slice is affected by a general system of forces. The thrust line connects the points of application of the interslice forces. The location of this thrust line may be assumed or its location may be determined using a rigorous method of analysis that satisfies complete equilibrium. 71

METHOD OF SLICES System of forces The popular simplified methods of analysis neglect the location of the interslice force because complete equilibrium is not satisfied for the failure mass. For this system, there are (6n 2) unknowns. Also, since only four equations for each slice (4n equations) can be written for the limit equilibrium for the system, the solution is statically indeterminate. 72

METHOD OF SLICES System of forces 73

METHOD OF SLICES Assumptions (3) A solution is possible providing the number of unknowns can be reduced by making some simplifying assumptions. One of these common assumptions is that the normal forces on the base of the slices acts at the midpoint, thus reducing the number of unknowns to (5n 2). This then requires an additional (n 2) assumptions to make the problem determinate. It is these assumptions that generally categorize the available methods of analysis. 74

METHOD OF SLICES Common Methods List of the common methods of analysis and the conditions of static equilibrium that are satisfied in determining the FOS. 75

METHOD OF SLICES General formulation Consider a body of soil on the point of sliding on the surface ABCD. For the purpose of analysis, we will divide the whole body of soil abode the surface of sliding into n elementary slices, separated by n-1 vertical boundaries. The choice of vertical interslice boundaries is merely a matter of convenience. In general, the failure condition is not satisfied on these surfaces. 76

METHOD OF SLICES General formulation If the body is stable, force and moment equilibrium conditions must be satisfied for each slice, and also for the whole body. The failure condition: τ must be satisfied everywhere on the surface ABCD. If F c = F φ = F along the failure surface, we may define a factor of safety in the form: f = c' + ( σ u) tanφ' n τ = τf F = c' F c + ( σ u) n tanφ' F φ 77

METHOD OF SLICES General formulation For the slice i: T i = [τ b sec α] i ; N i = [σ n b sec α] i Then: T i = ( τ b secα) = [ c' b secα + ( N u b secα) tanφ' ] i i 1 F It will be convenient to specify the pore pressure u i on BC in term of the pore pressure ratio r u equal to [u b/w] i with W i is the weight of slice i. T i = 1 F [ c' b secα + ( N W ru secα) tanφ' ] i (1) Vertical forces equilibrium of slice i where [T sin α + N cos α] i = [W - X] i X i = X (i+1) - X i (2) 78

METHOD OF SLICES General formulation Substituting equation (1) into number (2) we obtain Rearranging the terms, this yields (3) where Substitute equation (3) into number (1) to give (4) 79

METHOD OF SLICES General formulation Tangential forces equilibrium of slice i (5) Rearranging the terms, this yields Substituting T i with equation (4) we obtain In considering the equilibrium of the whole soil mass, the internal interslice forces (E 2 to E n and X 2 to X n ) must vanish. Also if there are no external forces on the end slices, E 1 = E (n+1) = X 1 = X (n+1) = 0 80

METHOD OF SLICES General formulation So that (6) Moment equilibrium of slice i around pivot O Since the moments of all internal forces (E i, X i ) must also vanish when considering the whole body, (7) 81

METHOD OF SLICES General formulation Substituting the expression for T i and N i obtained previously this becomes (8) In the case we have circular failure surface, equation (8) become (9) 82

METHOD OF SLICES OMS The Ordinary Method of Slices (OMS) (Fellenius, 1927, 1936) is one of the simplest procedures based on the method of slices and neglects all interslice forces (X i and E i ) and fails to satisfy force equilibrium for the slide mass as well as for individual slices. So the number of unknowns become: (5n 2) (n 1) (n 1) (n 1) = 2n +1 unknowns < 4n equations. X i E i z i This method consider a circular failure surface. If we remember that: and manipulating equation (7) to give 83

METHOD OF SLICES OMS we obtain The Ordinary Method of Slices can be used also in the case of layered soils. It is sufficient to consider the correct soil strength parameter (c and φ ) at the base of each slice. 84

METHOD OF SLICES Bishop s Method The Bishop s Method is one of the most famous methods based on limit equilibrium. He makes the hypothesis of a circular failure surface. There are two different approaches: a. Bishop s Rigorous Method. Satisfy complete equilibrium circular failure surface b. Bishop s Simplified Method. Satisfy moment equilibrium circular failure surface 85

METHOD OF SLICES Bishop s Method a. Bishop s Rigorous Method Bishop (1955) considers in his rigorous formulation equation (6) and equation (9). He also imposed that X i = λ f(x) where λ is a constant unknown and f(x) a known function. We saw early that there are n 2 additional unknown which make the problem indeterminate. In this case, due to the previous assumption, all X i become known if λ is defined. Therefore, the total number of unknowns is: (5n 2) (n 1) + 1 = 4n unknowns = 4n equation. The problem is determinate and the equilibrium equations are satisfied. 86

METHOD OF SLICES Bishop s Method Note: There could be several combinations of the unknown λ and the assumed known function f(x) that satisfy the problem. However, some of these combinations could be not admissible. It is important to check out other two requirements: 1. Shear strength along vertical interslice surfaces must be less than the failure strength of the soil, i.e. τ i =X i / A i < τ f = c + σ tan φ In fact, these surfaces can t reach the failure condition. 2. The thrust line must be located inside the sliding mass. 87

METHOD OF SLICES Bishop s Method b. Bishop s Simplified Method In his simplified approach, Bishop (1955) assumes that all vertical interslice shear forces X i are zero, reducing the number of unknowns by (n 1). This leaves (4n 1) unknowns, leaving the solution overdetermined. So, X i = 0, λ = 0 and X i = 0 give 88

METHOD OF SLICES Bishop s Method Solution procedure: 1) Assume initial value of F 0 2) Calculate with F 0 m α,i for each slice 3) Determine F with the above expression F 0 If F = F 0 : stop procedure If F F 0 : iterate procedure repeat steps (1) to (3) 89

METHOD OF SLICES Janbu s Method Similar to Bishop s method, the Janbu s method can be discerned in two different approaches: a. Janbu s Simplified Method. Satisfy force equilibrium Any kind of failure surface b. Janbu s Generalized Method. Janbu assumes a location of the thrust line, thereby reducing the number of unknowns to (4n 1). Similar to the rigorous Bishop method, Janbu also suggests that the actual location of the thrust line is an additional unknown, and thus equilibrium can be satisfied rigorously if the assumption selects the correct thrust line. Any kind of failure surface 90

METHOD OF SLICES Janbu s Method a. Janbu s Simplified Method Janbu (1954, 1957, 1973) assumes zero interslice vertical shear forces X i, reducing the number of unknowns to (4n 1). This leads to an overdetermined solution that will not satisfy moment equilibrium conditions. Equation (6) can be rewritten From which: 91

METHOD OF SLICES Janbu s Method The assumption of zero vertical interslice forces give an overvalued FOS. To account for this inadequacy, Janbu presented a correction factor, f 0 (>1), so that: This modification factor is a function of the slide geometry and the strength parameters of the soil. There is no consensus concerning the selection of the appropriate f 0 value for a surface intersecting different soil types. In cases where such a mixed variety of soils is present, the c φ curve is generally used to correct the calculated FOS value. 92

METHOD OF SLICES Janbu s Method For convenience, this modification factor can also be calculated according to the formula where b 1 varies according to the soil type: c only soils: b 1 = 0.69 φ only soils: b 1 = 0.31 c and φ soils: b 1 = 0.5 93

METHOD OF SLICES Advanced Methods Morgenstern Price Method Morgenstern and Price (1965) propose a method in which the inclination of the interslice resultant force is assumed to vary according to a portion of an arbitrary function, i.e. X i / E i = λ f(x) This additional portion of a selected function introduces an additional unknown, leaving 4n unknowns and 4n equations. The Morgenstern Price Method satisfy equations (6) and (8) and then the force and moment equilibrium for any kind of failure surface. Spencer s Method Spencer (1967, 1973) rigorously satisfies static equilibrium by assuming that the resultant interslice force has a constant, but unknown, inclination. This method derives from Morgenstern Price Method, assuming f(x) = cost. 94

METHOD OF SLICES GLE A general limit equilibrium (GLE) formulation (Chugh, 1986; Fredlund et al., 1981) can be developed to encompass most of the assumptions used by the various methods and may be used to analyze circular and noncircular failure surfaces. In view of this universal applicability, the GLE formulation has become one of the most popular methods as its generalization offers the ability to model a discrete version of the Morgenstern and Price (1965) procedure via the function used to describe the distribution of the interslice force angles. The method can be used to satisfy either force and moment equilibrium or, if required, just the force equilibrium conditions. 95

METHOD OF SLICES GLE The GLE procedure relies on the selection of an appropriate function that describes the equilibrium. variation of the interslice force angles to satisfy complete The main difficulty in using the GLE procedure is related to the requirement that the user verify the reliability and reasonableness of the calculated FOS. This additional complexity prevents the general use of the GLE method for automatic search procedures that attempt to identify the critical failure surface. However, single failure surfaces can be analyzed and the detailed solution examined for reasonableness. 96

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STABILITY CHARTS Slope stability charts are useful for preliminary analysis, to compare alternates that can later be examined by more detailed analyses. Chart solutions also provide a rapid means of checking the results of detailed analyses. Another use for slope stability charts is to back-calculate strength values for failed slopes to aid in planning remedial measures. This can be done by assuming an FOS of unity for the conditions at failure and solving for the unknown shear strength. The major shortcoming in using design charts is that most of them are for ideal, homogeneous soil conditions, which are not encountered in practice. 98

STABILITY CHARTS historical background 99

STABILITY CHARTS Taylor s Charts Taylor (1948) developed slope stability charts for soils with φ = 0 and φ > 0. As shown in these charts, the slope has an angle b, a height H, and base stratum at a depth of D H below the toe, where D is a depth ratio. The charts can be used to determine the developed cohesion, c d, as shown by the solid curves, and n H, which is the distance from the toe to the failure circle, as indicated by the short dashed curve. If there are loadings outside the toe that prevent the circle from passing below the toe, the long dashed curve should be used to determine the developed cohesion. Note that the solid and the long dashed curves converge as n approaches zero. The circle represented by the curves on the left of n = 0 do not pass below the toe, so the loading outside the toe has no influence on the developed cohesion. 100

STABILITY CHARTS Taylor s Charts 101

STABILITY CHARTS Taylor s Charts 102

STABILITY CHARTS Bishop-Morgenstern s Charts Bishop and Morgenstern s Procedure (1960) is quite more complex than Taylor s Charts. Compared to Taylor s method, this one, furthermore, can take into account water pore pressure inside the sliding mass and on the failure surface through water pore pressure ratio r u : Bishop and Morgenstern s Procedure is suitable for effective stress analysis in homogeneous soils. The charts are based on two parameters that are called stability factors m and n, so the FOS is defined as FOS = m n r u 103

STABILITY CHARTS Bishop-Morgenstern s Charts m and n depend on several parameters: β, slope angle; φ, soil friction angle; d f = H 1 /H, depth factor c / (γ H), similar to Taylor s stability number d f has a small influence on the solution, so one can refer essentially to three value of it: 1.00, 1.25, 1.50. In the charts are treated three conditions: c / (γ H) = 0.05 c / (γ H) = 0.025 c / (γ H) = 0 In general, user s condition is located between two of the previous situations. 104

STABILITY CHARTS Bishop-Morgenstern s Charts Normally, the user refers to an average r u (along failure surface) given by: where hi is the piezometric height of each of the slices in which the sliding mass can be subdivided. Note: This method doesn t give the critical surface, but gives information about d f. Given a series of geometrical and geotechnical parameters, it does exist a r u value indicated with r u,e, through FOS for d f = 1.0 is equal to FOS for d f = 1.25, so that: 105

STABILITY CHARTS Bishop-Morgenstern s Charts Analogously, it exists a value of r u which gives F 1.25 = F 1.5, that is If the calculated average r u is greater than r u,e, then F 1.25 < F 1, so the condition with d f = 1.25 is more critical. The same reasoning can be made comparing user s situation with d f = 1.25 condition and d f = 1.5 condition for establish which is the most critical situation. 106

STABILITY CHARTS Bishop-Morgenstern s Charts 107